m1une's library

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:heavy_check_mark: verify/graph/replacement_paths.test.cpp

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Code

#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"

#include <algorithm>
#include <cassert>
#include <functional>
#include "../../utilities/fast_io.hpp"
#include <limits>
#include <queue>
#include <random>
#include <utility>
#include <vector>

#include "../../graph/replacement_paths.hpp"

namespace {

using m1une::graph::Graph;
using m1une::graph::GraphPath;

template <class T>
T brute_distance(const Graph<T>& g, int s, int t, int removed_edge, int removed_vertex, T inf) {
    if (s == removed_vertex || t == removed_vertex) return inf;
    std::vector<T> dist(g.size(), inf);
    using P = std::pair<T, int>;
    std::priority_queue<P, std::vector<P>, std::greater<P>> que;
    dist[s] = T(0);
    que.emplace(T(0), s);
    while (!que.empty()) {
        auto [d, v] = que.top();
        que.pop();
        if (dist[v] != d) continue;
        for (const auto& e : g[v]) {
            if (!e.alive || e.id == removed_edge || e.to == removed_vertex) continue;
            T nd = d + e.cost;
            if (nd >= inf || dist[e.to] <= nd) continue;
            dist[e.to] = nd;
            que.emplace(nd, e.to);
        }
    }
    return dist[t];
}

template <class T>
void compare_with_brute(const Graph<T>& g, const GraphPath& path) {
    auto edge_result = m1une::graph::edge_replacement_paths(g, path);
    auto vertex_result = m1une::graph::vertex_replacement_paths(g, path);
    assert(edge_result.path.vertices == path.vertices);
    assert(edge_result.path.edges == path.edges);
    assert(vertex_result.path.vertices == path.vertices);
    assert(vertex_result.path.edges == path.edges);
    assert(edge_result.replacement_dist.size() == path.edges.size());
    assert(vertex_result.replacement_dist.size() == path.vertices.size());
    for (int i = 0; i < int(path.edges.size()); i++) {
        T expected = brute_distance(
            g, path.vertices.front(), path.vertices.back(), path.edges[i], -1, edge_result.inf
        );
        assert(edge_result.replacement_dist[i] == expected);
        assert(edge_result.reachable(i) == (expected != edge_result.inf));
    }
    for (int i = 0; i < int(path.vertices.size()); i++) {
        T expected = brute_distance(
            g, path.vertices.front(), path.vertices.back(), -1, path.vertices[i], vertex_result.inf
        );
        assert(vertex_result.replacement_dist[i] == expected);
        assert(vertex_result.reachable(i) == (expected != vertex_result.inf));
    }
}

template <class T>
void compare_automatic_with_brute(const Graph<T>& g, int s, int t) {
    auto edge_result = m1une::graph::edge_replacement_paths(g, s, t);
    auto vertex_result = m1une::graph::vertex_replacement_paths(g, s, t);
    assert(edge_result.path.vertices == vertex_result.path.vertices);
    assert(edge_result.path.edges == vertex_result.path.edges);
    compare_with_brute(g, edge_result.path);
    assert(edge_result.replacement_dist ==
           m1une::graph::edge_replacement_paths(g, edge_result.path).replacement_dist);
    assert(vertex_result.replacement_dist ==
           m1une::graph::vertex_replacement_paths(g, vertex_result.path).replacement_dist);
}

void test_only_path_and_trivial() {
    Graph<int> g(4);
    int e0 = g.add_edge(0, 1, 2);
    int e1 = g.add_edge(1, 2, 3);
    int e2 = g.add_edge(2, 3, 4);
    GraphPath path;
    path.vertices = {0, 1, 2, 3};
    path.edges = {e0, e1, e2};
    auto edges = m1une::graph::edge_replacement_paths(g, path);
    auto vertices = m1une::graph::vertex_replacement_paths(g, path);
    assert(std::all_of(edges.replacement_dist.begin(), edges.replacement_dist.end(),
                       [&](int x) { return x == edges.inf; }));
    assert(std::all_of(vertices.replacement_dist.begin(), vertices.replacement_dist.end(),
                       [&](int x) { return x == vertices.inf; }));
    compare_with_brute(g, path);

    auto trivial_edges = m1une::graph::edge_replacement_paths(g, 2, 2);
    auto trivial_vertices = m1une::graph::vertex_replacement_paths(g, 2, 2);
    assert((trivial_edges.path.vertices == std::vector<int>{2}));
    assert(trivial_edges.path.edges.empty());
    assert(trivial_edges.replacement_dist.empty());
    assert((trivial_vertices.replacement_dist == std::vector<int>{trivial_vertices.inf}));
}

void test_parallel_edges_and_inactive_edges() {
    Graph<long long> g(4);
    int fixed = g.add_edge(0, 1, 2);
    int equal_parallel = g.add_edge(0, 1, 2);
    int longer_parallel = g.add_edge(0, 1, 5);
    int e12 = g.add_edge(1, 2, 2);
    int e23 = g.add_edge(2, 3, 2);
    int inactive = g.add_edge(0, 3, 1);
    g.erase_edge(inactive);
    GraphPath path;
    path.vertices = {0, 1, 2, 3};
    path.edges = {fixed, e12, e23};
    auto result = m1une::graph::edge_replacement_paths(g, path);
    assert(result.replacement_dist[0] == 6);
    assert(equal_parallel != fixed && longer_parallel != fixed);
    compare_with_brute(g, path);
}

void test_detour_intervals_and_disconnections() {
    Graph<long long> g(9);
    int e0 = g.add_edge(0, 1, 1);
    int e1 = g.add_edge(1, 2, 1);
    int e2 = g.add_edge(2, 3, 1);
    int e3 = g.add_edge(3, 4, 1);
    g.add_edge(1, 5, 2);
    g.add_edge(5, 2, 2);
    g.add_edge(5, 3, 2);
    g.add_edge(0, 6, 4);
    g.add_edge(6, 3, 4);
    g.add_edge(1, 7, 3);
    g.add_edge(7, 4, 3);
    g.add_edge(0, 8, 20);
    g.add_edge(8, 4, 20);
    GraphPath path;
    path.vertices = {0, 1, 2, 3, 4};
    path.edges = {e0, e1, e2, e3};
    compare_with_brute(g, path);

    Graph<int> bridge(5);
    int b0 = bridge.add_edge(0, 1);
    int b1 = bridge.add_edge(1, 2);
    int b2 = bridge.add_edge(2, 3);
    int b3 = bridge.add_edge(3, 4);
    GraphPath bridge_path;
    bridge_path.vertices = {0, 1, 2, 3, 4};
    bridge_path.edges = {b0, b1, b2, b3};
    compare_with_brute(bridge, bridge_path);
}

void test_multiple_and_external_shortest_paths() {
    Graph<int> g(4);
    int e02 = g.add_edge(0, 2, 1);
    int e23 = g.add_edge(2, 3, 1);
    int e01 = g.add_edge(0, 1, 1);
    int e13 = g.add_edge(1, 3, 1);
    GraphPath through_two;
    through_two.vertices = {0, 2, 3};
    through_two.edges = {e02, e23};
    GraphPath through_one;
    through_one.vertices = {0, 1, 3};
    through_one.edges = {e01, e13};
    compare_with_brute(g, through_two);
    compare_with_brute(g, through_one);
    compare_automatic_with_brute(g, 0, 3);
    auto result = m1une::graph::edge_replacement_paths(g, through_two);
    assert(result.replacement_dist == std::vector<int>(2, 2));
}

template <class T>
std::vector<GraphPath> enumerate_shortest_paths(const Graph<T>& g, int s, int t, int limit) {
    auto from_s = m1une::graph::dijkstra(g, s);
    std::vector<GraphPath> paths;
    GraphPath current;
    current.vertices.push_back(s);
    std::vector<char> used(g.size(), false);
    used[s] = true;
    auto dfs = [&](auto&& self, int v) -> void {
        if (int(paths.size()) == limit) return;
        if (v == t) {
            paths.push_back(current);
            return;
        }
        for (const auto& e : g[v]) {
            if (!e.alive || used[e.to]) continue;
            if (from_s.dist[v] + e.cost != from_s.dist[e.to]) continue;
            used[e.to] = true;
            current.vertices.push_back(e.to);
            current.edges.push_back(e.id);
            self(self, e.to);
            current.edges.pop_back();
            current.vertices.pop_back();
            used[e.to] = false;
        }
    };
    dfs(dfs, s);
    return paths;
}

void test_randomized() {
    std::mt19937 rng(712367821);
    for (int iteration = 0; iteration < 1500; iteration++) {
        int n = 2 + int(rng() % 7);
        Graph<long long> g(n);
        int attempts = int(rng() % 22);
        for (int i = 0; i < attempts; i++) {
            int u = int(rng() % n);
            int v = int(rng() % n);
            if (u == v) continue;
            int id = g.add_edge(u, v, 1 + int(rng() % 10));
            if (rng() % 13 == 0) g.erase_edge(id);
        }
        int s = int(rng() % n);
        int t = int(rng() % n);
        if (s == t) t = (t + 1) % n;
        auto shortest = m1une::graph::dijkstra(g, s);
        if (!shortest.reachable(t)) continue;
        compare_automatic_with_brute(g, s, t);
        auto paths = enumerate_shortest_paths(g, s, t, 5);
        assert(!paths.empty());
        for (const auto& path : paths) compare_with_brute(g, path);
    }
}

}  // namespace

int main() {
    m1une::utilities::FastInput fast_input;
    m1une::utilities::FastOutput fast_output;

    test_only_path_and_trivial();
    test_parallel_edges_and_inactive_edges();
    test_detour_intervals_and_disconnections();
    test_multiple_and_external_shortest_paths();
    test_randomized();

    int a, b;
    fast_input >> a >> b;
    fast_output << a + b << '\n';
}
#line 1 "verify/graph/replacement_paths.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"

#include <algorithm>
#include <cassert>
#include <functional>
#line 1 "utilities/fast_io.hpp"



#line 5 "utilities/fast_io.hpp"
#include <array>
#include <cerrno>
#include <charconv>
#include <cstddef>
#include <cstdio>
#include <cstdlib>
#include <cstdint>
#include <cstring>
#include <iterator>
#include <string>
#include <sys/stat.h>
#include <type_traits>
#include <utility>
#include <unistd.h>
#include <vector>

namespace m1une {
namespace utilities {

struct FastOutput;

namespace internal {

// Shared with the convenience helpers in template.hpp.
inline FastOutput* standard_output_instance = nullptr;

// Detect std::begin(x), std::end(x).
template <class T, class = void>
struct is_range : std::false_type {};

template <class T>
struct is_range<T, std::void_t<
    decltype(std::begin(std::declval<T&>())),
    decltype(std::end(std::declval<T&>()))
>> : std::true_type {};

template <class T>
inline constexpr bool is_range_v = is_range<T>::value;

template <class T>
using range_reference_t = decltype(*std::begin(std::declval<T&>()));

template <class T>
using range_value_t = std::remove_cv_t<std::remove_reference_t<range_reference_t<T>>>;

template <class T, class = void>
struct range_stored_value {
    using type = range_value_t<T>;
};

template <class T>
struct range_stored_value<T, std::void_t<typename std::remove_cv_t<std::remove_reference_t<T>>::value_type>> {
    using type = typename std::remove_cv_t<std::remove_reference_t<T>>::value_type;
};

template <class T>
using range_stored_value_t = typename range_stored_value<T>::type;

// Treat strings and C strings as scalar output objects, not as ranges.
template <class T>
struct is_char_array : std::false_type {};

template <class T, std::size_t N>
struct is_char_array<T[N]>
    : std::bool_constant<std::is_same_v<std::remove_cv_t<T>, char>> {};

template <class T>
struct is_string_like
    : std::bool_constant<
          std::is_same_v<std::decay_t<T>, std::string>
          || std::is_same_v<std::decay_t<T>, const char*>
          || std::is_same_v<std::decay_t<T>, char*>
          || is_char_array<std::remove_reference_t<T>>::value
      > {};

template <class T>
inline constexpr bool is_string_like_v = is_string_like<T>::value;

// ModInt-like type: x.val() is printable, and x can be assigned from long long.
template <class T, class = void>
struct has_val_method : std::false_type {};

template <class T>
struct has_val_method<T, std::void_t<decltype(std::declval<const T&>().val())>>
    : std::true_type {};

template <class T>
inline constexpr bool has_val_method_v = has_val_method<T>::value;

template <class T, class = void>
struct has_static_mod_raw : std::false_type {};

template <class T>
struct has_static_mod_raw<
    T, std::void_t<decltype(T::mod()), decltype(T::raw(std::declval<uint32_t>()))>>
    : std::true_type {};

template <class T>
inline constexpr bool has_static_mod_raw_v = has_static_mod_raw<T>::value;

// libstdc++ before GCC 16 does not classify __int128 as an integral type in
// strict ISO modes such as -std=c++23. Keep the fast-I/O interface independent
// of that implementation detail.
template <class T>
inline constexpr bool is_integral_v =
    std::is_integral_v<T>
    || std::is_same_v<std::remove_cv_t<T>, __int128_t>
    || std::is_same_v<std::remove_cv_t<T>, __uint128_t>;

template <class T>
inline constexpr bool is_signed_v =
    std::is_signed_v<T>
    || std::is_same_v<std::remove_cv_t<T>, __int128_t>;

template <class T>
struct make_unsigned {
    using type = std::make_unsigned_t<T>;
};

template <>
struct make_unsigned<__int128_t> {
    using type = __uint128_t;
};

template <>
struct make_unsigned<__uint128_t> {
    using type = __uint128_t;
};

template <class T>
using make_unsigned_t = typename make_unsigned<std::remove_cv_t<T>>::type;

}  // namespace internal

struct FastInput {
    static constexpr int buffer_size = 1 << 20;

   private:
    std::FILE* _stream;
    char _buffer[buffer_size];
    int _position;
    int _length;
    int _file_descriptor;
    bool _streaming;

    bool refill() {
        _position = 0;
        if (_streaming) {
            ssize_t length;
            do {
                length = ::read(_file_descriptor, _buffer, buffer_size);
            } while (length < 0 && errno == EINTR);
            if (length <= 0) {
                _length = 0;
                return false;
            }
            _length = int(length);
        } else {
            _length = int(std::fread(_buffer, 1, buffer_size, _stream));
        }
        return _length != 0;
    }

    template <class T>
    bool read_integer_from_stream(T& value) {
        if (!skip_spaces()) return false;
        int c = read_char_raw();

        bool negative = false;
        if (c == '-') {
            negative = true;
            c = read_char_raw();
        }

        if constexpr (internal::is_signed_v<T>) {
            T result = 0;
            while ('0' <= c && c <= '9') {
                result = negative ? result * 10 - (c - '0')
                                  : result * 10 + (c - '0');
                c = read_char_raw();
            }
            value = result;
        } else {
            T result = 0;
            while ('0' <= c && c <= '9') {
                result = result * 10 + T(c - '0');
                c = read_char_raw();
            }
            value = negative ? T(0) - result : result;
        }
        return true;
    }

    bool prepare_number() {
        if (_length - _position >= 64) return true;
        const int remaining = _length - _position;
        if (remaining > 0) std::memmove(_buffer, _buffer + _position, remaining);
        const int added = int(std::fread(_buffer + remaining, 1, buffer_size - remaining, _stream));
        _position = 0;
        _length = remaining + added;
        if (_length < buffer_size) _buffer[_length] = '\0';
        return _length != 0;
    }

   public:
    explicit FastInput(std::FILE* stream = stdin)
        : _stream(stream),
          _position(0),
          _length(0),
          _file_descriptor(::fileno(stream)),
          _streaming([&] {
              struct stat status;
              return _file_descriptor >= 0
                     && ::fstat(_file_descriptor, &status) == 0
                     && !S_ISREG(status.st_mode);
          }()) {}

    FastInput(const FastInput&) = delete;
    FastInput& operator=(const FastInput&) = delete;

    int read_char_raw() {
        if (_position == _length && !refill()) return EOF;
        return _buffer[_position++];
    }

    bool skip_spaces() {
        int c = read_char_raw();
        while (c != EOF && c <= ' ') c = read_char_raw();
        if (c == EOF) return false;
        --_position;
        return true;
    }

    bool read(char& value) {
        if (!skip_spaces()) return false;
        value = char(read_char_raw());
        return true;
    }

    bool read(std::string& value) {
        if (!skip_spaces()) return false;
        value.clear();
        while (true) {
            const int begin = _position;
            while (_position < _length &&
                   static_cast<unsigned char>(_buffer[_position]) > ' ') {
                ++_position;
            }
            value.append(_buffer + begin, _position - begin);
            if (_position < _length) {
                ++_position;
                return true;
            }
            if (!refill()) return true;
        }
    }

    bool read(bool& value) {
        int x;
        if (!read(x)) return false;
        value = x != 0;
        return true;
    }

    template <class T>
    std::enable_if_t<
        internal::is_integral_v<T>
            && !std::is_same_v<std::remove_cv_t<T>, bool>
            && !std::is_same_v<std::remove_cv_t<T>, char>,
        bool
    >
    read(T& value) {
        if (_streaming) return read_integer_from_stream(value);
        if (!prepare_number()) return false;
        int c = static_cast<unsigned char>(_buffer[_position++]);
        while (c <= ' ') c = static_cast<unsigned char>(_buffer[_position++]);

        bool negative = false;
        if (c == '-') {
            negative = true;
            c = static_cast<unsigned char>(_buffer[_position++]);
        }

        if constexpr (internal::is_signed_v<T>) {
            T result = 0;
            while ('0' <= c && c <= '9') {
                const int first = c - '0';
                const int second = static_cast<unsigned char>(_buffer[_position]) - '0';
                if (0 <= second && second <= 9) {
                    result = negative ? result * 100 - (first * 10 + second)
                                      : result * 100 + (first * 10 + second);
                    ++_position;
                } else {
                    result = negative ? result * 10 - first : result * 10 + first;
                }
                c = static_cast<unsigned char>(_buffer[_position++]);
            }
            value = result;
        } else {
            T result = 0;
            while ('0' <= c && c <= '9') {
                const unsigned first = unsigned(c - '0');
                const int second = static_cast<unsigned char>(_buffer[_position]) - '0';
                if (0 <= second && second <= 9) {
                    result = result * 100 + T(first * 10 + unsigned(second));
                    ++_position;
                } else {
                    result = result * 10 + T(first);
                }
                c = static_cast<unsigned char>(_buffer[_position++]);
            }
            value = negative ? T(0) - result : result;
        }
        if (_position > _length) _position = _length;
        return true;
    }

    template <class T>
    std::enable_if_t<std::is_floating_point_v<T>, bool>
    read(T& value) {
        if (!skip_spaces()) return false;
        int c = read_char_raw();
        bool negative = false;
        if (c == '-' || c == '+') {
            negative = c == '-';
            c = read_char_raw();
        }

        long double result = 0;
        while ('0' <= c && c <= '9') {
            result = result * 10 + (c - '0');
            c = read_char_raw();
        }
        if (c == '.') {
            long double place = 0.1L;
            c = read_char_raw();
            while ('0' <= c && c <= '9') {
                result += (c - '0') * place;
                place *= 0.1L;
                c = read_char_raw();
            }
        }
        if (c == 'e' || c == 'E') {
            c = read_char_raw();
            bool exponent_negative = false;
            if (c == '-' || c == '+') {
                exponent_negative = c == '-';
                c = read_char_raw();
            }
            int exponent = 0;
            while ('0' <= c && c <= '9') {
                exponent = exponent * 10 + (c - '0');
                c = read_char_raw();
            }
            long double scale = 1;
            long double power = 10;
            while (exponent > 0) {
                if (exponent & 1) scale *= power;
                power *= power;
                exponent >>= 1;
            }
            result = exponent_negative ? result / scale : result * scale;
        }
        value = static_cast<T>(negative ? -result : result);
        return true;
    }

    template <class T>
    std::enable_if_t<
        internal::has_val_method_v<T>
            && !internal::is_integral_v<T>
            && !internal::is_range_v<T>,
        bool
    >
    read(T& value) {
        long long x;
        if (!read(x)) return false;
        if constexpr (internal::has_static_mod_raw_v<T>) {
            if (x >= 0 && uint64_t(x) < uint64_t(T::mod())) {
                value = T::raw(uint32_t(x));
            } else {
                value = T(x);
            }
        } else {
            value = T(x);
        }
        return true;
    }

    template <class First, class Second>
    bool read(std::pair<First, Second>& value) {
        if (!read(value.first)) return false;
        return read(value.second);
    }

    template <class Range>
    std::enable_if_t<
        internal::is_range_v<Range>
            && !internal::is_string_like_v<Range>,
        bool
    >
    read(Range& range) {
        using StoredValue = internal::range_stored_value_t<Range>;
        constexpr bool nested = internal::is_range_v<StoredValue>
                                && !internal::is_string_like_v<StoredValue>;

        for (auto&& value : range) {
            if constexpr (std::is_same_v<StoredValue, bool> && !nested) {
                bool x;
                if (!read(x)) return false;
                value = x;
            } else {
                if (!read(value)) return false;
            }
        }
        return true;
    }

    template <class First, class Second, class... Rest>
    bool read(First& first, Second& second, Rest&... rest) {
        if (!read(first)) return false;
        return read(second, rest...);
    }

    template <class T>
    FastInput& operator>>(T& value) {
        if (!read(value)) std::abort();
        return *this;
    }
};

struct FastOutput {
    static constexpr int buffer_size = 1 << 20;

   private:
    inline static const auto digit_quads = [] {
        std::array<char, 40000> result{};
        for (int i = 0; i < 10000; i++) {
            int value = i;
            for (int j = 3; j >= 0; j--) {
                result[4 * i + j] = char('0' + value % 10);
                value /= 10;
            }
        }
        return result;
    }();

    std::FILE* _stream;
    char _buffer[buffer_size];
    int _position;
    int _precision;
    std::chars_format _float_format;
    char _range_separator;
    std::string* _capture = nullptr;

    template <class T>
    std::string format_cell(const T& value) {
        std::string result;
        struct CaptureGuard {
            std::string*& target;
            std::string* previous;
            ~CaptureGuard() { target = previous; }
        } guard{_capture, _capture};
        _capture = &result;
        write(value);
        return result;
    }

    template <class Matrix>
    void write_aligned_matrix(const Matrix& matrix) {
        std::vector<std::vector<std::string>> rows;
        std::vector<std::size_t> widths;
        for (const auto& row : matrix) {
            auto& cells = rows.emplace_back();
            std::size_t column = 0;
            for (const auto& value : row) {
                cells.push_back(format_cell(value));
                if (column == widths.size()) widths.push_back(0);
                widths[column] = std::max(widths[column], cells.back().size());
                ++column;
            }
        }
        bool first = true;
        for (const auto& row : rows) {
            if (!first) write_char('\n');
            first = false;
            for (std::size_t column = 0; column < row.size(); ++column) {
                if (column != 0) write_char(_range_separator);
                for (std::size_t padding = row[column].size();
                     padding < widths[column]; ++padding) {
                    write_char(' ');
                }
                write(row[column]);
            }
        }
    }

   public:
    explicit FastOutput(std::FILE* stream = stdout)
        : _stream(stream),
          _position(0),
          _precision(6),
          _float_format(std::chars_format::general),
          _range_separator(' ') {
        if (_stream == stdout
            && internal::standard_output_instance == nullptr) {
            internal::standard_output_instance = this;
        }
    }

    FastOutput(const FastOutput&) = delete;
    FastOutput& operator=(const FastOutput&) = delete;

    ~FastOutput() {
        flush();
        if (internal::standard_output_instance == this) {
            internal::standard_output_instance = nullptr;
        }
    }

    void flush() {
        if (_position != 0) {
            std::fwrite(_buffer, 1, _position, _stream);
            _position = 0;
        }
        std::fflush(_stream);
    }

    void write_char(char c) {
        if (_capture != nullptr) {
            _capture->push_back(c);
            return;
        }
        if (_position == buffer_size) flush();
        _buffer[_position++] = c;
    }

    void write(const char* s) {
        while (*s != '\0') write_char(*s++);
    }

    void write(const std::string& s) {
        if (_capture != nullptr) {
            _capture->append(s);
            return;
        }
        std::size_t position = 0;
        while (position < s.size()) {
            if (_position == buffer_size) flush();
            const std::size_t copied =
                std::min<std::size_t>(buffer_size - _position, s.size() - position);
            std::memcpy(_buffer + _position, s.data() + position, copied);
            _position += int(copied);
            position += copied;
        }
    }

    void write(char c) {
        write_char(c);
    }

    void write(bool value) {
        write_char(value ? '1' : '0');
    }

    template <class T>
    std::enable_if_t<std::is_floating_point_v<T>>
    write(T value) {
        char digits[128];
        auto [end, error] = std::to_chars(
            digits,
            digits + sizeof(digits),
            value,
            _float_format,
            _precision
        );
        if (error != std::errc()) std::abort();
        for (const char* pointer = digits; pointer != end; pointer++) {
            write_char(*pointer);
        }
    }

    template <class T>
    std::enable_if_t<
        internal::is_integral_v<T>
            && !std::is_same_v<std::remove_cv_t<T>, bool>
            && !std::is_same_v<std::remove_cv_t<T>, char>
    >
    write(T value) {
        using Raw = std::remove_cv_t<T>;
        using Unsigned = internal::make_unsigned_t<Raw>;

        Unsigned magnitude;
        if constexpr (internal::is_signed_v<Raw>) {
            if (value < 0) {
                write_char('-');
                magnitude = Unsigned(0) - Unsigned(value);
            } else {
                magnitude = Unsigned(value);
            }
        } else {
            magnitude = value;
        }

        if (magnitude == 0) {
            write_char('0');
            return;
        }

        unsigned chunks[16];
        int count = 0;
        while (magnitude >= 10000) {
            const Unsigned quotient = magnitude / 10000;
            chunks[count++] = unsigned(magnitude - quotient * 10000);
            magnitude = quotient;
        }
        if (_capture == nullptr && _position > buffer_size - 64) flush();
        char captured[64];
        char* const begin = _capture != nullptr ? captured : _buffer + _position;
        char* destination = begin;
        const unsigned leading = unsigned(magnitude);
        const char* first = digit_quads.data() + 4 * leading;
        int skip = leading < 10 ? 3 : leading < 100 ? 2 : leading < 1000 ? 1 : 0;
        for (; skip < 4; skip++) *destination++ = first[skip];
        while (count--) {
            const char* digits = digit_quads.data() + 4 * chunks[count];
            std::memcpy(destination, digits, 4);
            destination += 4;
        }
        if (_capture != nullptr) {
            _capture->append(begin, destination - begin);
        } else {
            _position += int(destination - begin);
        }
    }

    template <class T>
    std::enable_if_t<
        internal::has_val_method_v<T>
            && !internal::is_integral_v<T>
            && !internal::is_range_v<T>
    >
    write(const T& value) {
        write(value.val());
    }

    template <class First, class Second>
    void write(const std::pair<First, Second>& value) {
        write(value.first);
        write_char(' ');
        write(value.second);
    }

    template <class Range>
    std::enable_if_t<
        internal::is_range_v<Range>
            && !internal::is_string_like_v<Range>
    >
    write(const Range& range) {
        using StoredValue = internal::range_stored_value_t<const Range>;
        constexpr bool nested = internal::is_range_v<StoredValue>
                                && !internal::is_string_like_v<StoredValue>;

        bool first = true;
        for (const auto& value : range) {
            if (!first) write_char(nested ? '\n' : _range_separator);
            first = false;
            if constexpr (std::is_same_v<StoredValue, bool> && !nested) {
                write(static_cast<bool>(value));
            } else {
                write(value);
            }
        }
    }

    template <class First, class... Rest>
    void print(const First& first, const Rest&... rest) {
        write(first);
        ((write_char(' '), write(rest)), ...);
    }

    void println() {
        write_char('\n');
    }

    void set_precision(int precision) {
        _precision = precision;
    }

    void set_fixed(int precision = 6) {
        _float_format = std::chars_format::fixed;
        _precision = precision;
    }

    void set_general(int precision = 6) {
        _float_format = std::chars_format::general;
        _precision = precision;
    }

    void set_range_separator(char separator) {
        _range_separator = separator;
    }

    template <class Matrix>
    void write_aligned(const Matrix& matrix) {
        using Row = internal::range_stored_value_t<const Matrix>;
        using Cell = internal::range_stored_value_t<const Row>;
        static_assert(internal::is_range_v<Row> && !internal::is_string_like_v<Row>,
                      "write_aligned requires a two-dimensional range");
        static_assert(!internal::is_range_v<Cell> || internal::is_string_like_v<Cell>,
                      "write_aligned requires scalar cells");
        write_aligned_matrix(matrix);
    }

    template <class Matrix>
    void println_aligned(const Matrix& matrix) {
        write_aligned(matrix);
        write_char('\n');
    }

    template <class... Args>
    void println(const Args&... args) {
        print(args...);
        write_char('\n');
    }

    template <class T>
    FastOutput& operator<<(const T& value) {
        write(value);
        return *this;
    }
};

}  // namespace utilities
}  // namespace m1une


#line 7 "verify/graph/replacement_paths.test.cpp"
#include <limits>
#include <queue>
#include <random>
#line 12 "verify/graph/replacement_paths.test.cpp"

#line 1 "graph/replacement_paths.hpp"



#line 11 "graph/replacement_paths.hpp"

#line 1 "graph/dijkstra.hpp"



#line 8 "graph/dijkstra.hpp"

#line 1 "graph/graph.hpp"



#line 8 "graph/graph.hpp"

namespace m1une {
namespace graph {

template <class T = int>
struct Edge {
    using cost_type = T;

    int from;
    int to;
    T cost;
    int id;
    bool alive;

    Edge() : from(-1), to(-1), cost(T()), id(-1), alive(true) {}
    Edge(int from_, int to_, T cost_ = T(1), int id_ = -1, bool alive_ = true)
        : from(from_), to(to_), cost(cost_), id(id_), alive(alive_) {}

    int other(int v) const {
        assert(v == from || v == to);
        return from ^ to ^ v;
    }
};

template <class T = int>
struct Graph {
    using edge_type = Edge<T>;
    using cost_type = T;

   private:
    struct EdgePositions {
        std::array<std::pair<int, int>, 2> value{};
        int size = 0;

        void push_back(std::pair<int, int> position) {
            assert(size < 2);
            value[size++] = position;
        }
    };

    int _n;
    int _edge_count;
    std::vector<std::vector<edge_type>> _g;
    std::vector<EdgePositions> _edge_positions;

   public:
    Graph() : _n(0), _edge_count(0) {}
    explicit Graph(int n) : _n(n), _edge_count(0), _g(n) {
        assert(0 <= n);
    }

    int size() const {
        return _n;
    }

    bool empty() const {
        return _n == 0;
    }

    int edge_count() const {
        return _edge_count;
    }

    int add_vertex() {
        _g.emplace_back();
        return _n++;
    }

    int add_directed_edge(int from, int to, T cost = T(1)) {
        assert(0 <= from && from < _n);
        assert(0 <= to && to < _n);
        int id = _edge_count++;
        int idx = int(_g[from].size());
        _g[from].push_back(edge_type(from, to, cost, id));
        _edge_positions.emplace_back();
        _edge_positions.back().push_back({from, idx});
        return id;
    }

    int add_edge(int u, int v, T cost = T(1)) {
        assert(0 <= u && u < _n);
        assert(0 <= v && v < _n);
        int id = _edge_count++;
        int u_idx = int(_g[u].size());
        _g[u].push_back(edge_type(u, v, cost, id));
        int v_idx = int(_g[v].size());
        _g[v].push_back(edge_type(v, u, cost, id));
        _edge_positions.emplace_back();
        _edge_positions.back().push_back({u, u_idx});
        _edge_positions.back().push_back({v, v_idx});
        return id;
    }

    void set_edge_alive(int id, bool alive) {
        assert(0 <= id && id < _edge_count);
        for (int i = 0; i < _edge_positions[id].size; ++i) {
            auto [v, idx] = _edge_positions[id].value[i];
            _g[v][idx].alive = alive;
        }
    }

    void erase_edge(int id) {
        set_edge_alive(id, false);
    }

    void revive_edge(int id) {
        set_edge_alive(id, true);
    }

    bool is_edge_alive(int id) const {
        assert(0 <= id && id < _edge_count);
        assert(_edge_positions[id].size != 0);
        auto [v, idx] = _edge_positions[id].value[0];
        return _g[v][idx].alive;
    }

    const std::vector<edge_type>& operator[](int v) const {
        assert(0 <= v && v < _n);
        return _g[v];
    }

    std::vector<edge_type>& operator[](int v) {
        assert(0 <= v && v < _n);
        return _g[v];
    }

    const std::vector<std::vector<edge_type>>& adjacency() const {
        return _g;
    }

    std::vector<std::vector<edge_type>>& adjacency() {
        return _g;
    }

    std::vector<edge_type> edges(bool include_inactive = false) const {
        std::vector<edge_type> result;
        result.reserve(_edge_count);
        std::vector<char> used(_edge_count, false);
        for (int v = 0; v < _n; v++) {
            for (const auto& e : _g[v]) {
                if (!include_inactive && !e.alive) continue;
                if (0 <= e.id && e.id < _edge_count) {
                    if (used[e.id]) continue;
                    used[e.id] = true;
                }
                result.push_back(e);
            }
        }
        return result;
    }

    Graph reversed() const {
        Graph result(_n);
        result._edge_count = _edge_count;
        result._edge_positions.assign(_edge_count, {});
        for (int v = 0; v < _n; v++) {
            for (const auto& e : _g[v]) {
                int idx = int(result._g[e.to].size());
                result._g[e.to].push_back(edge_type(e.to, e.from, e.cost, e.id, e.alive));
                if (0 <= e.id && e.id < _edge_count) result._edge_positions[e.id].push_back({e.to, idx});
            }
        }
        return result;
    }
};

}  // namespace graph
}  // namespace m1une


#line 10 "graph/dijkstra.hpp"

namespace m1une {
namespace graph {

template <class T>
struct DijkstraResult {
    std::vector<T> dist;
    std::vector<char> reached;
    std::vector<int> parent;
    std::vector<int> parent_edge;
    T inf = T();

    bool reachable(int v) const {
        assert(0 <= v && v < int(dist.size()));
        return reached[v];
    }

    std::vector<int> path(int t) const {
        assert(reachable(t));
        std::vector<int> result;
        for (int v = t; v != -1; v = parent[v]) result.push_back(v);
        std::reverse(result.begin(), result.end());
        return result;
    }
};

namespace internal {

template <class T>
class DijkstraHeap {
   private:
    const std::vector<T>& dist_;
    std::vector<int> heap_;
    std::vector<int> position_;

    bool less(int first, int second) const {
        return dist_[heap_[first]] < dist_[heap_[second]];
    }

    void swap_nodes(int first, int second) {
        std::swap(heap_[first], heap_[second]);
        position_[heap_[first]] = first;
        position_[heap_[second]] = second;
    }

    void sift_up(int index) {
        while (index != 0) {
            const int parent = (index - 1) / 2;
            if (!less(index, parent)) break;
            swap_nodes(index, parent);
            index = parent;
        }
    }

    void sift_down(int index) {
        while (2 * index + 1 < int(heap_.size())) {
            int child = 2 * index + 1;
            if (child + 1 < int(heap_.size()) && less(child + 1, child)) {
                ++child;
            }
            if (!less(child, index)) break;
            swap_nodes(index, child);
            index = child;
        }
    }

   public:
    DijkstraHeap(const std::vector<T>& dist, int size)
        : dist_(dist), position_(size, -1) {
        heap_.reserve(size);
    }

    bool empty() const {
        return heap_.empty();
    }

    void push_or_decrease(int vertex) {
        int& position = position_[vertex];
        if (position == -1) {
            position = int(heap_.size());
            heap_.push_back(vertex);
        }
        sift_up(position);
    }

    int pop_min() {
        const int result = heap_.front();
        position_[result] = -1;
        if (heap_.size() == 1) {
            heap_.pop_back();
            return result;
        }
        heap_.front() = heap_.back();
        position_[heap_.front()] = 0;
        heap_.pop_back();
        sift_down(0);
        return result;
    }
};

}  // namespace internal

template <class T>
DijkstraResult<T> dijkstra(const Graph<T>& g,
                           const std::vector<int>& sources) {
    int n = g.size();
    DijkstraResult<T> result;
    result.dist.resize(n);
    result.reached.assign(n, false);
    result.parent.assign(n, -1);
    result.parent_edge.assign(n, -1);

    internal::DijkstraHeap<T> que(result.dist, n);
    for (int s : sources) {
        assert(0 <= s && s < n);
        if (result.reached[s]) continue;
        result.reached[s] = true;
        result.dist[s] = T();
        que.push_or_decrease(s);
    }

    while (!que.empty()) {
        const int current = que.pop_min();
        for (const auto& e : g[current]) {
            if (!e.alive) continue;
            T nd = result.dist[current] + e.cost;
            if (result.reached[e.to] && !(nd < result.dist[e.to])) continue;
            result.reached[e.to] = true;
            result.dist[e.to] = std::move(nd);
            result.parent[e.to] = current;
            result.parent_edge[e.to] = e.id;
            que.push_or_decrease(e.to);
        }
    }

    return result;
}

template <class T>
DijkstraResult<T> dijkstra(const Graph<T>& g, int s) {
    return dijkstra(g, std::vector<int>{s});
}

// Compatibility overload: unreachable distances are replaced by inf after the
// search. Reachability itself never depends on this sentinel.
template <class T>
DijkstraResult<T> dijkstra(const Graph<T>& g,
                           const std::vector<int>& sources, const T& inf) {
    DijkstraResult<T> result = dijkstra(g, sources);
    result.inf = inf;
    for (int v = 0; v < int(result.dist.size()); v++) {
        if (!result.reachable(v)) result.dist[v] = inf;
    }
    return result;
}

template <class T>
DijkstraResult<T> dijkstra(const Graph<T>& g, int s, const T& inf) {
    return dijkstra(g, std::vector<int>{s}, inf);
}

}  // namespace graph
}  // namespace m1une


#line 14 "graph/replacement_paths.hpp"

namespace m1une {
namespace graph {

struct GraphPath {
    std::vector<int> vertices;
    std::vector<int> edges;
};

template <class T>
struct EdgeReplacementPathsResult {
    GraphPath path;
    std::vector<T> replacement_dist;
    T inf;

    bool reachable(int path_edge_index) const {
        assert(0 <= path_edge_index && path_edge_index < int(replacement_dist.size()));
        return replacement_dist[path_edge_index] != inf;
    }
};

template <class T>
struct VertexReplacementPathsResult {
    GraphPath path;
    std::vector<T> replacement_dist;
    T inf;

    bool reachable(int path_vertex_index) const {
        assert(0 <= path_vertex_index && path_vertex_index < int(replacement_dist.size()));
        return replacement_dist[path_vertex_index] != inf;
    }
};

namespace internal {

template <class T>
T replacement_paths_safe_add(T a, T b, T inf) {
    if (a >= inf || b >= inf) return inf;
    if (a > inf - b) return inf;
    return a + b;
}

template <class T>
DijkstraResult<T> replacement_paths_dijkstra(const Graph<T>& g, int s, T inf) {
    int n = g.size();
    assert(0 <= s && s < n);
    DijkstraResult<T> result;
    result.dist.assign(n, inf);
    result.reached.assign(n, false);
    result.parent.assign(n, -1);
    result.parent_edge.assign(n, -1);
    result.inf = inf;

    using P = std::pair<T, int>;
    std::priority_queue<P, std::vector<P>, std::greater<P>> que;
    result.dist[s] = T(0);
    result.reached[s] = true;
    que.emplace(T(0), s);
    while (!que.empty()) {
        auto [d, v] = que.top();
        que.pop();
        if (result.dist[v] != d) continue;
        for (const auto& e : g[v]) {
            if (!e.alive) continue;
            T nd = replacement_paths_safe_add(d, e.cost, inf);
            if (result.dist[e.to] <= nd) continue;
            result.reached[e.to] = true;
            result.dist[e.to] = nd;
            result.parent[e.to] = v;
            result.parent_edge[e.to] = e.id;
            que.emplace(nd, e.to);
        }
    }
    return result;
}

template <class T>
std::vector<Edge<T>> replacement_paths_validate_graph(const Graph<T>& g, T inf) {
    assert(T(0) < inf);
    std::vector<int> occurrence(g.edge_count(), 0);
    std::vector<Edge<T>> edge_by_id(g.edge_count());
    for (int v = 0; v < g.size(); v++) {
        for (const auto& e : g[v]) {
            assert(e.from == v);
            assert(0 <= e.to && e.to < g.size());
            assert(0 <= e.id && e.id < g.edge_count());
            if (e.alive) assert(T(0) < e.cost);
            if (occurrence[e.id] == 0) {
                edge_by_id[e.id] = e;
            } else {
                assert(occurrence[e.id] == 1);
                const auto& other = edge_by_id[e.id];
                assert(e.from == other.to && e.to == other.from);
                assert(e.cost == other.cost && e.alive == other.alive);
            }
            occurrence[e.id]++;
        }
    }

    for (int id = 0; id < g.edge_count(); id++) {
        // add_edge creates exactly two mutually reversed arcs with one logical id.
        assert(occurrence[id] == 2);
    }
    return edge_by_id;
}

template <class T>
void replacement_paths_validate_path(const Graph<T>& g, const GraphPath& path,
                                     const std::vector<Edge<T>>& edge_by_id,
                                     const DijkstraResult<T>& from_s, T inf) {
    assert(!path.vertices.empty());
    assert(path.edges.size() + 1 == path.vertices.size());
    std::vector<char> used_vertex(g.size(), false);
    for (int v : path.vertices) {
        assert(0 <= v && v < g.size());
        assert(!used_vertex[v]);
        used_vertex[v] = true;
    }

    T path_cost = T(0);
    for (int i = 0; i < int(path.edges.size()); i++) {
        int id = path.edges[i];
        assert(0 <= id && id < g.edge_count());
        assert(g.is_edge_alive(id));
        const auto& e = edge_by_id[id];
        int u = path.vertices[i];
        int v = path.vertices[i + 1];
        assert((e.from == u && e.to == v) || (e.from == v && e.to == u));
        assert(T(0) < e.cost);
        path_cost = replacement_paths_safe_add(path_cost, e.cost, inf);
    }
    assert(from_s.reachable(path.vertices.back()));
    assert(path_cost == from_s.dist[path.vertices.back()]);
}

template <class T>
GraphPath replacement_paths_restore_path(const DijkstraResult<T>& result, int s, int t) {
    assert(result.reachable(t));
    GraphPath path;
    for (int v = t; v != s; v = result.parent[v]) {
        assert(v != -1 && result.parent[v] != -1 && result.parent_edge[v] != -1);
        path.vertices.push_back(v);
        path.edges.push_back(result.parent_edge[v]);
    }
    path.vertices.push_back(s);
    std::reverse(path.vertices.begin(), path.vertices.end());
    std::reverse(path.edges.begin(), path.edges.end());
    return path;
}

template <class T>
struct ReplacementPathsData {
    GraphPath path;
    std::vector<T> dist_s;
    std::vector<T> dist_t;
    std::vector<int> block;
    std::vector<char> is_path_edge;
    std::vector<Edge<T>> edge_by_id;
    T inf;
};

template <class T>
ReplacementPathsData<T> replacement_paths_prepare(const Graph<T>& g, const GraphPath& path,
                                                   T inf, const DijkstraResult<T>* known_from_s) {
    auto edge_by_id = replacement_paths_validate_graph(g, inf);
    int s = path.vertices.front();
    int t = path.vertices.back();
    auto computed_from_s = known_from_s == nullptr
                               ? replacement_paths_dijkstra(g, s, inf)
                               : DijkstraResult<T>();
    const auto& from_s = known_from_s == nullptr ? computed_from_s : *known_from_s;
    replacement_paths_validate_path(g, path, edge_by_id, from_s, inf);
    auto from_t = replacement_paths_dijkstra(g, t, inf);

    int n = g.size();
    std::vector<int> path_position(n, -1);
    std::vector<char> is_path_edge(g.edge_count(), false);
    for (int i = 0; i < int(path.vertices.size()); i++) path_position[path.vertices[i]] = i;
    for (int id : path.edges) is_path_edge[id] = true;

    std::vector<int> parent(n, -1);
    for (int i = 0; i < int(path.edges.size()); i++) {
        int v = path.vertices[i + 1];
        parent[v] = path.vertices[i];
        const auto& e = edge_by_id[path.edges[i]];
        assert(replacement_paths_safe_add(from_s.dist[parent[v]], e.cost, inf) == from_s.dist[v]);
    }
    for (int v = 0; v < n; v++) {
        if (!from_s.reachable(v) || v == s || path_position[v] != -1) continue;
        for (const auto& e : g[v]) {
            if (!e.alive || !from_s.reachable(e.to)) continue;
            if (replacement_paths_safe_add(from_s.dist[e.to], e.cost, inf) != from_s.dist[v]) {
                continue;
            }
            parent[v] = e.to;
            break;
        }
        assert(parent[v] != -1);
        assert(from_s.dist[parent[v]] < from_s.dist[v]);
    }

    std::vector<std::vector<int>> children(n);
    for (int v = 0; v < n; v++) {
        if (parent[v] != -1) children[parent[v]].push_back(v);
    }
    std::vector<int> block(n, -1);
    block[s] = 0;
    std::vector<int> stack = {s};
    while (!stack.empty()) {
        int v = stack.back();
        stack.pop_back();
        for (int to : children[v]) {
            block[to] = path_position[to] == -1 ? block[v] : path_position[to];
            stack.push_back(to);
        }
    }
    for (int v = 0; v < n; v++) assert(!from_s.reachable(v) || block[v] != -1);

    return {path, from_s.dist, from_t.dist, block, is_path_edge, edge_by_id, inf};
}

template <class T>
class ReplacementPathsRangeChmin {
   private:
    int _size;
    std::vector<T> _lazy;

   public:
    ReplacementPathsRangeChmin(int n, T inf) : _size(1) {
        while (_size < n) _size <<= 1;
        _lazy.assign(2 * _size, inf);
    }

    void apply(int l, int r, T value) {
        assert(0 <= l && l <= r && r <= _size);
        for (l += _size, r += _size; l < r; l >>= 1, r >>= 1) {
            if (l & 1) {
                _lazy[l] = std::min(_lazy[l], value);
                l++;
            }
            if (r & 1) {
                --r;
                _lazy[r] = std::min(_lazy[r], value);
            }
        }
    }

    std::vector<T> values(int n) {
        for (int v = 1; v < _size; v++) {
            _lazy[2 * v] = std::min(_lazy[2 * v], _lazy[v]);
            _lazy[2 * v + 1] = std::min(_lazy[2 * v + 1], _lazy[v]);
        }
        return std::vector<T>(_lazy.begin() + _size, _lazy.begin() + _size + n);
    }
};

template <class T>
std::vector<T> replacement_paths_solve_edges(const ReplacementPathsData<T>& data) {
    int answer_size = int(data.path.edges.size());
    ReplacementPathsRangeChmin<T> range_chmin(answer_size, data.inf);
    for (const auto& e : data.edge_by_id) {
        if (!e.alive || data.is_path_edge[e.id]) continue;
        int u = e.from;
        int v = e.to;
        if (data.block[u] == -1 || data.block[v] == -1 || data.block[u] == data.block[v]) continue;
        if (data.block[u] > data.block[v]) std::swap(u, v);
        int a = data.block[u];
        int b = data.block[v];
        T candidate = replacement_paths_safe_add(data.dist_s[u], e.cost, data.inf);
        candidate = replacement_paths_safe_add(candidate, data.dist_t[v], data.inf);
        if (candidate == data.inf) continue;
        range_chmin.apply(a, b, candidate);
    }
    return range_chmin.values(answer_size);
}

template <class T>
T replacement_paths_without_vertex(const Graph<T>& g, int s, int t, int removed, T inf) {
    if (s == removed || t == removed) return inf;
    std::vector<T> dist(g.size(), inf);
    using P = std::pair<T, int>;
    std::priority_queue<P, std::vector<P>, std::greater<P>> que;
    dist[s] = T(0);
    que.emplace(T(0), s);
    while (!que.empty()) {
        auto [d, v] = que.top();
        que.pop();
        if (dist[v] != d) continue;
        for (const auto& e : g[v]) {
            if (!e.alive || e.to == removed) continue;
            T nd = replacement_paths_safe_add(d, e.cost, inf);
            if (dist[e.to] <= nd) continue;
            dist[e.to] = nd;
            que.emplace(nd, e.to);
        }
    }
    return dist[t];
}

template <class T>
std::vector<T> replacement_paths_solve_vertices(const Graph<T>& g,
                                                const ReplacementPathsData<T>& data) {
    // One edge can cross an edge cut, but a vertex-avoiding path may enter and
    // leave the failed vertex's tree block through two different detour edges.
    int path_size = int(data.path.vertices.size());
    std::vector<T> answer(path_size, data.inf);
    int s = data.path.vertices.front();
    int t = data.path.vertices.back();
    for (int i = 1; i + 1 < path_size; i++) {
        answer[i] = replacement_paths_without_vertex(g, s, t, data.path.vertices[i], data.inf);
    }
    return answer;
}

}  // namespace internal

template <class T>
EdgeReplacementPathsResult<T> edge_replacement_paths(
    const Graph<T>& g, const GraphPath& path, T inf = std::numeric_limits<T>::max() / T(4)) {
    assert(!path.vertices.empty());
    auto data = internal::replacement_paths_prepare(
        g, path, inf, static_cast<const DijkstraResult<T>*>(nullptr));
    auto replacement_dist = internal::replacement_paths_solve_edges(data);
    return {path, replacement_dist, inf};
}

template <class T>
EdgeReplacementPathsResult<T> edge_replacement_paths(
    const Graph<T>& g, int s, int t, T inf = std::numeric_limits<T>::max() / T(4)) {
    assert(0 <= s && s < g.size());
    assert(0 <= t && t < g.size());
    auto from_s = internal::replacement_paths_dijkstra(g, s, inf);
    assert(from_s.reachable(t));
    auto path = internal::replacement_paths_restore_path(from_s, s, t);
    auto data = internal::replacement_paths_prepare(g, path, inf, &from_s);
    auto replacement_dist = internal::replacement_paths_solve_edges(data);
    return {path, replacement_dist, inf};
}

template <class T>
VertexReplacementPathsResult<T> vertex_replacement_paths(
    const Graph<T>& g, const GraphPath& path, T inf = std::numeric_limits<T>::max() / T(4)) {
    assert(!path.vertices.empty());
    auto data = internal::replacement_paths_prepare(
        g, path, inf, static_cast<const DijkstraResult<T>*>(nullptr));
    auto replacement_dist = internal::replacement_paths_solve_vertices(g, data);
    return {path, replacement_dist, inf};
}

template <class T>
VertexReplacementPathsResult<T> vertex_replacement_paths(
    const Graph<T>& g, int s, int t, T inf = std::numeric_limits<T>::max() / T(4)) {
    assert(0 <= s && s < g.size());
    assert(0 <= t && t < g.size());
    auto from_s = internal::replacement_paths_dijkstra(g, s, inf);
    assert(from_s.reachable(t));
    auto path = internal::replacement_paths_restore_path(from_s, s, t);
    auto data = internal::replacement_paths_prepare(g, path, inf, &from_s);
    auto replacement_dist = internal::replacement_paths_solve_vertices(g, data);
    return {path, replacement_dist, inf};
}

}  // namespace graph
}  // namespace m1une


#line 14 "verify/graph/replacement_paths.test.cpp"

namespace {

using m1une::graph::Graph;
using m1une::graph::GraphPath;

template <class T>
T brute_distance(const Graph<T>& g, int s, int t, int removed_edge, int removed_vertex, T inf) {
    if (s == removed_vertex || t == removed_vertex) return inf;
    std::vector<T> dist(g.size(), inf);
    using P = std::pair<T, int>;
    std::priority_queue<P, std::vector<P>, std::greater<P>> que;
    dist[s] = T(0);
    que.emplace(T(0), s);
    while (!que.empty()) {
        auto [d, v] = que.top();
        que.pop();
        if (dist[v] != d) continue;
        for (const auto& e : g[v]) {
            if (!e.alive || e.id == removed_edge || e.to == removed_vertex) continue;
            T nd = d + e.cost;
            if (nd >= inf || dist[e.to] <= nd) continue;
            dist[e.to] = nd;
            que.emplace(nd, e.to);
        }
    }
    return dist[t];
}

template <class T>
void compare_with_brute(const Graph<T>& g, const GraphPath& path) {
    auto edge_result = m1une::graph::edge_replacement_paths(g, path);
    auto vertex_result = m1une::graph::vertex_replacement_paths(g, path);
    assert(edge_result.path.vertices == path.vertices);
    assert(edge_result.path.edges == path.edges);
    assert(vertex_result.path.vertices == path.vertices);
    assert(vertex_result.path.edges == path.edges);
    assert(edge_result.replacement_dist.size() == path.edges.size());
    assert(vertex_result.replacement_dist.size() == path.vertices.size());
    for (int i = 0; i < int(path.edges.size()); i++) {
        T expected = brute_distance(
            g, path.vertices.front(), path.vertices.back(), path.edges[i], -1, edge_result.inf
        );
        assert(edge_result.replacement_dist[i] == expected);
        assert(edge_result.reachable(i) == (expected != edge_result.inf));
    }
    for (int i = 0; i < int(path.vertices.size()); i++) {
        T expected = brute_distance(
            g, path.vertices.front(), path.vertices.back(), -1, path.vertices[i], vertex_result.inf
        );
        assert(vertex_result.replacement_dist[i] == expected);
        assert(vertex_result.reachable(i) == (expected != vertex_result.inf));
    }
}

template <class T>
void compare_automatic_with_brute(const Graph<T>& g, int s, int t) {
    auto edge_result = m1une::graph::edge_replacement_paths(g, s, t);
    auto vertex_result = m1une::graph::vertex_replacement_paths(g, s, t);
    assert(edge_result.path.vertices == vertex_result.path.vertices);
    assert(edge_result.path.edges == vertex_result.path.edges);
    compare_with_brute(g, edge_result.path);
    assert(edge_result.replacement_dist ==
           m1une::graph::edge_replacement_paths(g, edge_result.path).replacement_dist);
    assert(vertex_result.replacement_dist ==
           m1une::graph::vertex_replacement_paths(g, vertex_result.path).replacement_dist);
}

void test_only_path_and_trivial() {
    Graph<int> g(4);
    int e0 = g.add_edge(0, 1, 2);
    int e1 = g.add_edge(1, 2, 3);
    int e2 = g.add_edge(2, 3, 4);
    GraphPath path;
    path.vertices = {0, 1, 2, 3};
    path.edges = {e0, e1, e2};
    auto edges = m1une::graph::edge_replacement_paths(g, path);
    auto vertices = m1une::graph::vertex_replacement_paths(g, path);
    assert(std::all_of(edges.replacement_dist.begin(), edges.replacement_dist.end(),
                       [&](int x) { return x == edges.inf; }));
    assert(std::all_of(vertices.replacement_dist.begin(), vertices.replacement_dist.end(),
                       [&](int x) { return x == vertices.inf; }));
    compare_with_brute(g, path);

    auto trivial_edges = m1une::graph::edge_replacement_paths(g, 2, 2);
    auto trivial_vertices = m1une::graph::vertex_replacement_paths(g, 2, 2);
    assert((trivial_edges.path.vertices == std::vector<int>{2}));
    assert(trivial_edges.path.edges.empty());
    assert(trivial_edges.replacement_dist.empty());
    assert((trivial_vertices.replacement_dist == std::vector<int>{trivial_vertices.inf}));
}

void test_parallel_edges_and_inactive_edges() {
    Graph<long long> g(4);
    int fixed = g.add_edge(0, 1, 2);
    int equal_parallel = g.add_edge(0, 1, 2);
    int longer_parallel = g.add_edge(0, 1, 5);
    int e12 = g.add_edge(1, 2, 2);
    int e23 = g.add_edge(2, 3, 2);
    int inactive = g.add_edge(0, 3, 1);
    g.erase_edge(inactive);
    GraphPath path;
    path.vertices = {0, 1, 2, 3};
    path.edges = {fixed, e12, e23};
    auto result = m1une::graph::edge_replacement_paths(g, path);
    assert(result.replacement_dist[0] == 6);
    assert(equal_parallel != fixed && longer_parallel != fixed);
    compare_with_brute(g, path);
}

void test_detour_intervals_and_disconnections() {
    Graph<long long> g(9);
    int e0 = g.add_edge(0, 1, 1);
    int e1 = g.add_edge(1, 2, 1);
    int e2 = g.add_edge(2, 3, 1);
    int e3 = g.add_edge(3, 4, 1);
    g.add_edge(1, 5, 2);
    g.add_edge(5, 2, 2);
    g.add_edge(5, 3, 2);
    g.add_edge(0, 6, 4);
    g.add_edge(6, 3, 4);
    g.add_edge(1, 7, 3);
    g.add_edge(7, 4, 3);
    g.add_edge(0, 8, 20);
    g.add_edge(8, 4, 20);
    GraphPath path;
    path.vertices = {0, 1, 2, 3, 4};
    path.edges = {e0, e1, e2, e3};
    compare_with_brute(g, path);

    Graph<int> bridge(5);
    int b0 = bridge.add_edge(0, 1);
    int b1 = bridge.add_edge(1, 2);
    int b2 = bridge.add_edge(2, 3);
    int b3 = bridge.add_edge(3, 4);
    GraphPath bridge_path;
    bridge_path.vertices = {0, 1, 2, 3, 4};
    bridge_path.edges = {b0, b1, b2, b3};
    compare_with_brute(bridge, bridge_path);
}

void test_multiple_and_external_shortest_paths() {
    Graph<int> g(4);
    int e02 = g.add_edge(0, 2, 1);
    int e23 = g.add_edge(2, 3, 1);
    int e01 = g.add_edge(0, 1, 1);
    int e13 = g.add_edge(1, 3, 1);
    GraphPath through_two;
    through_two.vertices = {0, 2, 3};
    through_two.edges = {e02, e23};
    GraphPath through_one;
    through_one.vertices = {0, 1, 3};
    through_one.edges = {e01, e13};
    compare_with_brute(g, through_two);
    compare_with_brute(g, through_one);
    compare_automatic_with_brute(g, 0, 3);
    auto result = m1une::graph::edge_replacement_paths(g, through_two);
    assert(result.replacement_dist == std::vector<int>(2, 2));
}

template <class T>
std::vector<GraphPath> enumerate_shortest_paths(const Graph<T>& g, int s, int t, int limit) {
    auto from_s = m1une::graph::dijkstra(g, s);
    std::vector<GraphPath> paths;
    GraphPath current;
    current.vertices.push_back(s);
    std::vector<char> used(g.size(), false);
    used[s] = true;
    auto dfs = [&](auto&& self, int v) -> void {
        if (int(paths.size()) == limit) return;
        if (v == t) {
            paths.push_back(current);
            return;
        }
        for (const auto& e : g[v]) {
            if (!e.alive || used[e.to]) continue;
            if (from_s.dist[v] + e.cost != from_s.dist[e.to]) continue;
            used[e.to] = true;
            current.vertices.push_back(e.to);
            current.edges.push_back(e.id);
            self(self, e.to);
            current.edges.pop_back();
            current.vertices.pop_back();
            used[e.to] = false;
        }
    };
    dfs(dfs, s);
    return paths;
}

void test_randomized() {
    std::mt19937 rng(712367821);
    for (int iteration = 0; iteration < 1500; iteration++) {
        int n = 2 + int(rng() % 7);
        Graph<long long> g(n);
        int attempts = int(rng() % 22);
        for (int i = 0; i < attempts; i++) {
            int u = int(rng() % n);
            int v = int(rng() % n);
            if (u == v) continue;
            int id = g.add_edge(u, v, 1 + int(rng() % 10));
            if (rng() % 13 == 0) g.erase_edge(id);
        }
        int s = int(rng() % n);
        int t = int(rng() % n);
        if (s == t) t = (t + 1) % n;
        auto shortest = m1une::graph::dijkstra(g, s);
        if (!shortest.reachable(t)) continue;
        compare_automatic_with_brute(g, s, t);
        auto paths = enumerate_shortest_paths(g, s, t, 5);
        assert(!paths.empty());
        for (const auto& path : paths) compare_with_brute(g, path);
    }
}

}  // namespace

int main() {
    m1une::utilities::FastInput fast_input;
    m1une::utilities::FastOutput fast_output;

    test_only_path_and_trivial();
    test_parallel_edges_and_inactive_edges();
    test_detour_intervals_and_disconnections();
    test_multiple_and_external_shortest_paths();
    test_randomized();

    int a, b;
    fast_input >> a >> b;
    fast_output << a + b << '\n';
}
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